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nullhomotopy -- make a null homotopy

Description

nullhomotopy f -- produce a nullhomotopy for a map f of chain complexes.

Whether f is null homotopic is not checked.

Here is part of an example provided by Luchezar Avramov. We construct a random module over a complete intersection, resolve it over the polynomial ring, and produce a null homotopy for the map that is multiplication by one of the defining equations for the complete intersection.

i1 : A = ZZ/101[x,y];
i2 : M = cokernel random(A^3, A^{-2,-2})

o2 = cokernel | 2x2+40xy-22y2  9x2-20xy-31y2   |
              | 46x2+25xy+33y2 35x2+16xy+44y2  |
              | x2-39xy-46y2   -29x2-23xy-11y2 |

                            3
o2 : A-module, quotient of A
i3 : R = cokernel matrix {{x^3,y^4}}

o3 = cokernel | x3 y4 |

                            1
o3 : A-module, quotient of A
i4 : N = prune (M**R)

o4 = cokernel | 3x2+42xy-39y2 -35x2+27xy-46y2 x3 x2y-3xy2    8xy2-17y3   y4 0  0  |
              | x2-3xy-20y2   -39xy+44y2      0  -38xy2-12y3 -21xy2+30y3 0  y4 0  |
              | -17xy-15y2    x2+36xy+38y2    0  -41y3       xy2+20y3    0  0  y4 |

                            3
o4 : A-module, quotient of A
i5 : C = resolution N

      3      8      5
o5 = A  <-- A  <-- A  <-- 0
                           
     0      1      2      3

o5 : ChainComplex
i6 : d = C.dd

          3                                                                             8
o6 = 0 : A  <------------------------------------------------------------------------- A  : 1
               | 3x2+42xy-39y2 -35x2+27xy-46y2 x3 x2y-3xy2    8xy2-17y3   y4 0  0  |
               | x2-3xy-20y2   -39xy+44y2      0  -38xy2-12y3 -21xy2+30y3 0  y4 0  |
               | -17xy-15y2    x2+36xy+38y2    0  -41y3       xy2+20y3    0  0  y4 |

          8                                                                             5
     1 : A  <------------------------------------------------------------------------- A  : 2
               {2} | 14xy2+20y3     -44xy2+26y3    -14y3      25y3      31y3       |
               {2} | -31xy2+19y3    -50y3          31y3       6y3       -21y3      |
               {3} | 46xy-19y2      49xy+3y2       -46y2      4y2       48y2       |
               {3} | -46x2+21xy+5y2 -49x2-18xy+8y2 46xy-2y2   -4xy-31y2 -48xy-29y2 |
               {3} | 31x2+41xy-26y2 20xy+40y2      -31xy+41y2 -6xy-37y2 21xy+5y2   |
               {4} | 0              0              x-39y      16y       25y        |
               {4} | 0              0              31y        x-36y     36y        |
               {4} | 0              0              33y        20y       x-26y      |

          5
     2 : A  <----- 0 : 3
               0

o6 : ChainComplexMap
i7 : s = nullhomotopy (x^3 * id_C)

          8                            3
o7 = 1 : A  <------------------------ A  : 0
               {2} | 0 x+3y 39y   |
               {2} | 0 17y  x-36y |
               {3} | 1 -3   35    |
               {3} | 0 39   10    |
               {3} | 0 -41  2     |
               {4} | 0 0    0     |
               {4} | 0 0    0     |
               {4} | 0 0    0     |

          5                                                                              8
     2 : A  <-------------------------------------------------------------------------- A  : 1
               {5} | -28 26 0 5y      -13x-21y xy-32y2      23xy+17y2    -30xy+22y2 |
               {5} | -43 50 0 -33x-7y 39x+6y   38y2         xy+44y2      21xy+7y2   |
               {5} | 0   0  0 0       0        x2+39xy+14y2 -16xy+7y2    -25xy-39y2 |
               {5} | 0   0  0 0       0        -31xy-26y2   x2+36xy-13y2 -36xy-43y2 |
               {5} | 0   0  0 0       0        -33xy-10y2   -20xy-5y2    x2+26xy-y2 |

                   5
     3 : 0 <----- A  : 2
              0

o7 : ChainComplexMap
i8 : s*d + d*s

          3                    3
o8 = 0 : A  <---------------- A  : 0
               | x3 0  0  |
               | 0  x3 0  |
               | 0  0  x3 |

          8                                       8
     1 : A  <----------------------------------- A  : 1
               {2} | x3 0  0  0  0  0  0  0  |
               {2} | 0  x3 0  0  0  0  0  0  |
               {3} | 0  0  x3 0  0  0  0  0  |
               {3} | 0  0  0  x3 0  0  0  0  |
               {3} | 0  0  0  0  x3 0  0  0  |
               {4} | 0  0  0  0  0  x3 0  0  |
               {4} | 0  0  0  0  0  0  x3 0  |
               {4} | 0  0  0  0  0  0  0  x3 |

          5                              5
     2 : A  <-------------------------- A  : 2
               {5} | x3 0  0  0  0  |
               {5} | 0  x3 0  0  0  |
               {5} | 0  0  x3 0  0  |
               {5} | 0  0  0  x3 0  |
               {5} | 0  0  0  0  x3 |

     3 : 0 <----- 0 : 3
              0

o8 : ChainComplexMap
i9 : s^2

          5         3
o9 = 2 : A  <----- A  : 0
               0

                   8
     3 : 0 <----- A  : 1
              0

o9 : ChainComplexMap

Ways to use nullhomotopy :